Your favorite paradox?

[quatona]

i dunno if someone has already said this but, my fav paradox is the paradox of a stone " if God is all powerful, could he make a stone so heavy he cannot lift" i have many others but this one will always be my fav ( i dnt give my personal responses)

Don´t say that! You are still so young and there are so many way more beautiful paradoxes out there waiting for you to discover them.

 

[Lifesaver]

I love the "surprise test" paradox.

The Logic Professor tells his students: "Next week there will be a surprise test. By this I mean that you will not know the day of the test before the day arrives."

The cleverest student then thought to himself: "Well, if the test is to be a surprise, then it can't fall on Friday. Because if we reach Thursday and the test has still not been applied, then we'll know that it will be on Friday, and thus it won't be a surprise test anymore, as we'll know its day before the day has arrived."

This thought led him to another one: "If it can't be on Friday, it can't be on Thursday as well! For if we get to Wednesday not having had the test, and we know that it can't be on Friday, then the only option left is Thursday; and thus we'll know on Wednesday that the test will fall on Thursday, and thus it won't be a surprise test."

Proceeding in this manner, the student ruled out all days. Afterall, the same argument works for Wednesday and Tuesday, leaving only Monday. But if there is only Monday as a possibility, then he knows the test will be on Monday. And this it is no longer a surprise test. Therefore, it can't be on Monday either.

"Therefore, there will be no surprise test at all! The Professor is just playing a game on us!"

He tells his perfect reasoning to his classmates. Finding no flaw in it, they are all convinced and go home for the weekend, during which none of them studied anything, confident the surprise test was just a game.

The next week begins, and on Tuesday the Professor arrives with the test-sheets and announces the surprise test. All the students become very concerned.

The cleverest one, who had dismissed the surprise test before others, tells the Professor why he can't possibly be applying the surprise test, since no matter which day he chose, it could not be a surprise.

But the Professor answers: "If that were true, you would not be surprised right now, would you?"

And therefore, by logically proving that there was no day in which the surprise test could be applied the student has to accept that the surprise test can be applied on any day.

 

[TricksterWolf]

I love the "surprise test" paradox.

The Logic Professor tells his students: "Next week there will be a surprise test. By this I mean that you will not know the day of the test before the day arrives."

The cleverest student then thought to himself: "Well, if the test is to be a surprise, then it can't fall on Friday. Because if we reach Thursday and the test has still not been applied, then we'll know that it will be on Friday, and thus it won't be a surprise test anymore, as we'll know its day before the day has arrived."

This thought led him to another one: "If it can't be on Friday, it can't be on Thursday as well! For if we get to Wednesday not having had the test, and we know that it can't be on Friday, then the only option left is Thursday; and thus we'll know on Wednesday that the test will fall on Thursday, and thus it won't be a surprise test."

Proceeding in this manner, the student ruled out all days. Afterall, the same argument works for Wednesday and Tuesday, leaving only Monday. But if there is only Monday as a possibility, then he knows the test will be on Monday. And this it is no longer a surprise test. Therefore, it can't be on Monday either.

"Therefore, there will be no surprise test at all! The Professor is just playing a game on us!"

He tells his perfect reasoning to his classmates. Finding no flaw in it, they are all convinced and go home for the weekend, during which none of them studied anything, confident the surprise test was just a game.

The next week begins, and on Tuesday the Professor arrives with the test-sheets and announces the surprise test. All the students become very concerned.

The cleverest one, who had dismissed the surprise test before others, tells the Professor why he can't possibly be applying the surprise test, since no matter which day he chose, it could not be a surprise.

But the Professor answers: "If that were true, you would not be surprised right now, would you?"

And therefore, by logically proving that there was no day in which the surprise test could be applied the student has to accept that the surprise test can be applied on any day.

The logic the students used is only true if there exists no possibility that the suprise test may not occur at all. By believing that this is an additional possibility, as the students all did, their entire argument is invalidated!

Booyah.

Trickster

 

[Lifesaver]

But indeed, there was no possibility of the non-occurrence of the test. The Professor meant it for real and he never lies!

 

[TricksterWolf]

But indeed, there was no possibility of the non-occurrence of the test. The Professor meant it for real and he never lies!

That statement is in error, though, if it comes in conjunction with the original story. If that is the case, the students are simply surprised because the professor lied.

Trickster

 

[Lifesaver]

It is not in error at all.
The Professor told them there would be a surprise test, and there was a surprise test! He spoke only the truth.

The Professor did not lie, and the students were surprised because the Professor applied the test in a day which they did not expect.

 

[TricksterWolf]

It is not in error at all.
The Professor told them there would be a surprise test, and there was a surprise test! He spoke only the truth.

The Professor did not lie, and the students were surprised because the Professor applied the test in a day which they did not expect.

No. The Professor did not lie; but he also did not tell the truth!

The Professor told them that they would have a test, and that they would not know the day of the test until the test arrived. By strict reverse induction, this statement is neither true nor false (like "this statement is a lie"). Since a statement that is neither classified as true nor false is "not true", it contradicts the assertion that "the Professor always tells the truth". The reason the students were surprised was that the Professor did not tell the truth, in the strictest logical sense.

Trickster

 

[Pesto]

Ceci n'est pas une pipe.

 

[Lifesaver]

I don't understand your objection, Tricksterwolf. What the Professor said was true: they would have a surprise test and they would not know its day until the day arrived.
That is exactly what happened.

 

[TricksterWolf]

I don't understand your objection, Tricksterwolf. What the Professor said was true: they would have a surprise test and they would not know its day until the day arrived.
That is exactly what happened.

Consider a two-day week. The Professor says that the students will have a surprise test, and they won't know the day until the day arrives. Obviously, if the test is given on Tuesday, the Professor is a liar--the students will know on Monday when there is no test. This means that if the Professor tells the truth, he must give the test on Monday. But this is also impossible, since it is the only remaining option, and the Professor is still a liar; whether the students are "surprised" is insufficient to make the Professor's statement true. The fact that the students don't expect the test is simply an indication that the students based their reasoning on the perception that the Professor was telling the truth in a strict logical fashion. He wasn't.

The same situation exists in the five-day condition. Yes, the students are unprepared for the test; this is because the Professor lied to them when he gave them an unfullfillable test condition. It isn't possible to have a surprise test at all according to the definitions he gave, in the strict logical sense.

The reason that this doesn't make rational sense when we say it two ways is that people don't speak in logical, mathematically provable statements. When someone says "there will be a surprise test and you won't know the day before it happens" this indicates a general condition of "surprise" to us, without the mathematical underpinnings. Certainly, there are several cases that seem equally unpredictable to us, since we're not machines: Monday, Tuesday, Wednesday. But each of these cases is predictable enough that, strictly speaking, it is not allowed by the definition the teacher originally gave. You're mixing psychological meaning with mathematical, hence the confusion.

Trickster

 

[Lifesaver]

Of course he was telling the truth.
He said "you will be surprised". They were surprised. He spoke strictly the truth, in the 5 or in the 2-say week.

The students reasoned exactly as you did: the Professor can't be speaking the truth, they concluded; it's only a game.
But this very conclusion turns in favour of the Professor, because now he can give the test in any day of the week, and the students will be surprised.

If the students didn't try to figure out the day, they would have been surprised.
By trying to figure the day out and realizing that the Professor cannot possibly surprise them, they in fact make it so that any day will be a surprise.

In either case the Professor surprises them. He is telling the truth.

Psychology plays absolutely no part in this answer. "Being surprised" means not knowing the day of the test until the day has arrived.

 

[TricksterWolf]

Of course he was telling the truth.
He said "you will be surprised". They were surprised. He spoke strictly the truth, in the 5 or in the 2-say week.

The students reasoned exactly as you did: the Professor can't be speaking the truth, they concluded; it's only a game.
But this very conclusion turns in favour of the Professor, because now he can give the test in any day of the week, and the students will be surprised.

If the students didn't try to figure out the day, they would have been surprised.
By trying to figure the day out and realizing that the Professor cannot possibly surprise them, they in fact make it so that any day will be a surprise.

In either case the Professor surprises them. He is telling the truth.

Psychology plays absolutely no part in this answer. "Being surprised" means not knowing the day of the test until the day has arrived.

He doesn't only say "you will be surprised". You're eliminating the paradoxical elements when it suits you.

Trickster